How do you test the alternating series Sigma (-1)^(n+1)(1-1/n) from n is [1,oo) for convergence?

1 Answer
Nov 7, 2017

See below.

Explanation:

sum_(k=1)^oo(-1)^k a_k = sum_(k=1)^ooa_(2k-1)-a_(2k) = sum_(k=1)^oo b_k

but

b_k = 1-1/(2k-1)-(1-1/(2k)) = -1/((2k)(2k-1))

so we conclude that sum_(k=1)^oo b_k converges

NOTE:

sum_(k=1)^oo 1/(2k(2k-1)) le sum_(k=1)^oo 1/(2k)^2 = 1/4 pi^2/6

(See Basel problem https://en.wikipedia.org/wiki/Basel_problem )